(57h) The Elasto-Capillary Landau-Levich Coating Flow Problem

This paper concerns interfacial flows of complex fluids, a subject to which Gary Leal has made significant contributions. We consider the dip-coating flow problem when the interface has both an elastic bending stiffness and a constant surface tension.  When the elasticity is absent, the classical analysis of Landau & Levich predicts a unique coating thickness as a function of the capillary number of withdrawal.  We consider both a purely elastic problem when surface tension is absent, and an elasto-capillary problem when both elasticity and surface tension are present.  In the case where interfacial tension is negligible, we assume the elasticity number El – the ratio of surface elasticity to viscous forces - is small and develop the solution for the free boundary as a matched asymptotic expansion in the small parameter El^1/7, thus determining the film thickness as a function of El.  A remarkable aspect of the problem is the occurrence of multiple solutions, and five of these are found numerically. In any event, the film thickness varies as El^4/7, or equivalently, U^4/7, where U is the plate speed, in agreement with previous experiments.  The solution for the elasto-capillary problem is formulated in a similar way, with an elasto-capillary number, e, (the ratio of elasticity to surface tension), as an additional parameter. It is shown that it is possible to connect the problems of pure elasticity and elasto-capillarity respectively through the parameter e, but that connecting one of the five elasto-capillary branches to the classical result remains an elusive goal.